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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.00302 |
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Table of Contents:
- For a relational structure ${\mathbb X}$ we investigate the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X}):=\{ f[X]: f\in \mathop{\rm Emb}\nolimits ({\mathbb X})\}$. Here we consider uncountable ordinals. Since $\mathop{\rm sq}\nolimits {\mathbb P} (α)$ is isomorphic to the direct product $\prod _{i=1}^n (\mathop{\rm sq}\nolimits {\mathbb P} (ω^{δ_i}))^{s_i}$, where $α= ω^{δ_n}s_n+\dots +ω^{δ_1}s_1+ m$ is the Cantor normal form for $α$, the analysis is reduced to the investigation of the posets of the form ${\mathbb P} (ω^{δ})$. It turns out that, in ZFC, either the poset $\mathop{\rm sq}\nolimits {\mathbb P} (α)$ is $σ$-closed and completely embeds $P(ω)/\mathop{\rm Fin}$ and, hence, preserves $ω_1$ and forces $|{\mathfrak c}|=|{\mathfrak h}|$, or, otherwise, completely embeds the algebra $P(λ)/[λ]^{<λ}$, for some regular $ω<λ\leq \mathop{\rm cf}\nolimits (δ)$, and collapses $ω_2$ to $ω$. Regarding the Cantor normal form, the first case appears iff for each $i\leq n$ we have $\mathop{\rm cf}\nolimits (δ_i)\leq ω$, or $δ_i = θ_i + \mathop{\rm cf}\nolimits (δ_i )$, where $\mathop{\mathrm{Ord}}\nolimits \niθ_i \geq \mathop{\rm cf}\nolimits (δ_i ) >\mathop{\rm cf}\nolimits (θ_i )=ω$ and $θ_i =\lim _{n\rightarrow ω}δ_n$, where $\mathop{\rm cf}\nolimits (δ_n)=\mathop{\rm cf}\nolimits (δ_i)$, for all $n\in ω$.